src/HOL/Probability/Borel_Space.thy
changeset 50021 d96a3f468203
parent 50003 8c213922ed49
child 50087 635d73673b5e
equal deleted inserted replaced
50020:6b9611abcd4c 50021:d96a3f468203
    72 proof (rule measurable_measure_of, simp_all)
    72 proof (rule measurable_measure_of, simp_all)
    73   fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M"
    73   fix S :: "'x set" assume "open S" thus "f -` S \<inter> space M \<in> sets M"
    74     using assms[of S] by simp
    74     using assms[of S] by simp
    75 qed
    75 qed
    76 
    76 
    77 lemma borel_measurable_const[measurable (raw)]:
    77 lemma borel_measurable_const:
    78   "(\<lambda>x. c) \<in> borel_measurable M"
    78   "(\<lambda>x. c) \<in> borel_measurable M"
    79   by auto
    79   by auto
    80 
    80 
    81 lemma borel_measurable_indicator:
    81 lemma borel_measurable_indicator:
    82   assumes A: "A \<in> sets M"
    82   assumes A: "A \<in> sets M"
   166   assumes f[measurable]: "f \<in> borel_measurable M"
   166   assumes f[measurable]: "f \<in> borel_measurable M"
   167   assumes g[measurable]: "g \<in> borel_measurable M"
   167   assumes g[measurable]: "g \<in> borel_measurable M"
   168   shows borel_measurable_le[measurable]: "{w \<in> space M. f w \<le> g w} \<in> sets M"
   168   shows borel_measurable_le[measurable]: "{w \<in> space M. f w \<le> g w} \<in> sets M"
   169     and borel_measurable_eq[measurable]: "{w \<in> space M. f w = g w} \<in> sets M"
   169     and borel_measurable_eq[measurable]: "{w \<in> space M. f w = g w} \<in> sets M"
   170     and borel_measurable_neq: "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
   170     and borel_measurable_neq: "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
   171   unfolding eq_iff not_less[symmetric] by measurable+
   171   unfolding eq_iff not_less[symmetric]
       
   172   by measurable
   172 
   173 
   173 subsection "Borel space equals sigma algebras over intervals"
   174 subsection "Borel space equals sigma algebras over intervals"
   174 
   175 
   175 lemma rational_boxes:
   176 lemma rational_boxes:
   176   fixes x :: "'a\<Colon>ordered_euclidean_space"
   177   fixes x :: "'a\<Colon>ordered_euclidean_space"